#### Date of Award

2022

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Graduate Group

Mathematics

#### First Advisor

Robin Pemantle

#### Abstract

Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of generating function coefficients in a direction r. It uses Morse theory on the pole variety V := {H = 0} ⊆ (C∗)d to deform the torus T in the multivariate Cauchy Integral Formula via the downward gradient flow for the log-linear function h = hr = − ∑ rj log |zj|, giving a homology decomposition of T into cycles around critical points of h on V . The deformation can flow to infinity at finite height when the height function is not a proper map. This happens only in the presence of a critical point at infinity (CPAI): a sequence of points on V approaching a point at infinity, and such that log-normals to V converge projectively to r. The CPAI is called heighted if the height function also converges to a finite value. The central questions that I have attempted to answer involve analyzing whether all CPAI are heighted, and in which directions CPAI can occur. I attempted to answer these questions by examining sequences converging to faces of the toric compactification defined by a multiple of the Newton polytope P of the polynomial H. The idea is to show that any projective limit of log-normals of a sequence converging to a face F must be parallel to F. This turns out to be true but only under further hypotheses. It implies that CPAI must always be heighted and can only occur in directions parallel to some face of P. The extra hypotheses hold in smooth cases under generically satisfied conditions. In addition, I show under a smoothness condition, that a point in a codimension-1 face F can only be a CPAI for directions parallel to F, and that the directions for a codimension-2 face can be a larger set, which can be computed explicitly and still has positive codimension. The question of whether non-heighted CPAI exist in general is left open; I conjecture that they do not exist.

#### Recommended Citation

Gillen, Stephen, "Geometry Of Gradient Flows For Analytic Combinatorics" (2022). *Publicly Accessible Penn Dissertations*. 4904.

https://repository.upenn.edu/edissertations/4904